Solving Variational Inequalities Adaptive Projection Algorithm

Variational inequality problems arise in a widely fields such as signal processing, system identification, filter design, robot control, economic science, transportation science, operational research, Management, physics, nonlinear analysis fields and so on. Especially, many problems in scientific and engineering fields such as various kinds of obstacles problems, earth dam seepage problems, elastoplasticity contact problems and ice block melt problems, and mathematical programming, complementary problem and fixed point problem, can be formulated into variational inequality problem. Thus variational inequality problems provide a uniform framework for solving many optimization problems. Therefore, how to solve variational inequality problem effectively has important theory and application values.Over last decades, Many numerical methods have been proposed including proximal point algorithms, alternating directions methods, Newton-type methods and projection methods, ect., Among them, the projection methods are attractive for their simplicity and efficiency, when the projection can be easily computed. Even though some efforts have been made in solving variational inequality problem by projection methods, the exitsting projection methods have slow convergence and strong convergent condition. So, two new self adaptive projection methods are proposed on the basis of the existing projection methods, and their global convergence are proved strictly. New methods have overcome the shortcoming of the existing projection methods. The efficiency of the new methods aer illustrated bu some preliminary computational results.The thesis is divided into four parts. The main results of chapters as following:The first part is preliminaries. We introduce some definitions, significance and development of the variational inequality, projection theory, some properties of projection operator and somebasic theory of variational inequality are introduced. Moreover, some classic methods and theirproperties, and the development of the projection methods are studied.In the second part, based on the exsiting projection methods, a new self-adaptive projection methods for variational inequalities is proposed. The new method has improved the direction and step-size of the existing projection methods. The new direction and step-size are not zero near the solution such that the new method has quick convergence. we also have shown that the global convergence of the new method under the pesudomotonicity of the underlying mapping, which overcome the strong convergent condition of the existing methods. Because of employing self adaptive strategy, the convergence of the new method has nothing to do with the parameters. Meanwhile, its linear convergence is obtained. The efficiency of the new method is illustrated by some preliminary computational results. In the third part, a new self adaptive projection method for variational inequality is proposed. Compared with the method in the second part, the new method uses different step strategy. It has been proved that the new step is not zero near the solution, which make the method is quick convergence. Meanwhile, its global convergence and the convergence speed are analyzed under the pseudomotonicity of the underlying mapping. The efficience of the new method is illustrated by some preliminary computational results.The last part is a conclusion about the main work in the thesis, and we will do further research in variational inequality from several aspects .